PDEs

Apply d Alembert formula to solve the following problem Question : Let u(x,t) be the solution of the initial value problem \[u_{tt}-u_{xx}=0\]\[u\left(x,0\right)=x^3,\ u_t\left(x,0\right)=\sin(x)\]Then \[u\left(\pi,\pi\right)\] is1. \[4\pi^3\]2. \[\pi^3\]3. \[0\]4. \[0\] Solution…

Question : What is the solution of the Cauchy Problem pde \[u_t-uu_x=0\ ,\ \ x\in R,\ t>0\] \[u\left(x,0\right)=x,\ \ x\in R\] Solution : For the given Cauchy Problem pde\[u_t-uu_x=0\ ,\…

We shall Apply separation of variables Method to solve heat equation

Question : Find the solution of pde 3d^2z/dx^2+2d^2z/dxdy-5d^2z/dy^2=0 Solution : Given PDE [3\frac{\partial^2z}{\partial x^2}+2\frac{\partial^2z}{\partial x\partial y}-5\frac{\partial^2z}{\partial y^2}=0 ] Is a Homogeneous Linear PDE with constant coefficients. To solve such equations…

find solution of heat equation ut=3uxx with Dirichlet boundary conditions Solution : \[u_t=3u_{xx}\ ,\ 0<x<2\ ,\ t>0\] (1) Boundary conditions \[u\left(0,\ t\right)=0,\ \ u\left(2,\ t\right)=0\ \ \ for\ all\ t>0…

Question : how to Solve ut = kuxx (−∞ < x < ∞, 0 < t < ∞)  u(x, 0) = e^(3x) Solution : To solve \[u_t\ \ =\ ku_{xx}\…

Question : solve the wave equation utt = c2 uxx for −∞ < x < +∞  with the initial conditions u(x, 0) = φ(x)= sin x , ut(x, 0) =…

Question : Find general solution of the pde ut + ux = 0 Solution :

Question : how to determine first order pdes as linear homogeneous linear inhomogeneous or nonlinear Solution :